On the Zeros of Certain Linear Combinations of Poincaré Cusp Forms

Poincaré series are of fundamental importance in the theory of modular forms. For a fixed even integer k ≥ 4 , the space of modular forms of weight k can be spanned by certain Poincaré series. We provide a criterion for a modular form to have a specified number of zeros on the unit circle in the standard fundamental domain for the action of SL(2,Z) on the complex upper half-plane. The criterion is given in terms of the coefficients that are obtained by writing the modular form as a linear combination of the generating Poincaré series.